The Arithmetic of Digit Polygons: Spectral Orthogonality, Carry Defects, and the Geometry of Reciprocal Prime Addition

This paper studies the geometry of adding reciprocal primes through the lens of their digit polygons. For primes p and q coprime to the base, the decimal addition 1/p + 1/q = (p+q) / (pq) produces a digit polygon whose signed area is analyzed via an exact three-layer decomposition. The area addition law decomposes the carry-free (pointwise) sum's area into individual areas plus a cross-term equal to the polarization of the shoelace quadratic form. The spectral orthogonality theorem establishes that this cross-area vanishes if and only if the multiplicative orders of the base mod p and mod q have gcd at most 2, so digit polygons of coprime-period primes are geometrically non-interacting under addition; when the periods share a larger common factor the cross-area is controlled by Gauss sum estimates. The carry defect — the difference between the actual area and the carry-free area — is proved to be expressible as an exact quadratic function of the carry vector and is shown to be positive (area-reducing) for coprime-period pairs with sufficiently large primes, meaning carries smooth the polygon toward its centroid. Applied iteratively to the cumulative sum SN = Σ 1/pᵢ, the universal attractor theorem proves that the normalized area A (SN) /k (SN) converges to −33/8 regardless of which primes are summed, with super-exponentially growing period and diverging total area. The Mertens polygon correspondence translates Mertens' theorem on the sum of reciprocal primes into a geometric statement: integer crossings of SN produce transient polygon degeneracies that resolve within O (1) additional primes as equidistribution reasserts itself.